Closed-form evaluations of Fibonacci Lucas reciprocal sums with three factors

Transcription

1 Notes on Nuber Theory Discrete Matheatics Print ISSN Online ISSN Vol No Closed-for evaluations of Fibonacci Lucas reciprocal sus with three factors Robert Frontczak Lesbank Baden-Wuertteberg A Hauptbahnhof Stuttgart Gerany e-ail: robert.frontczak@lbbw.de Received: 3 October 206 Accepted: 3 March 207 Abstract: In this article we present expressions for certain types of reciprocal Fibonacci Lucas sus. The coon feature of the sus is that in each case the denoinator of the su consists of a product of three Fibonacci or Lucas nubers. Keywords: Fibonacci nuber Lucas nuber Reciprocal su. AMS Classification: B37 B39. Introduction Generalized Fibonacci nubers G n ay be defined through the second order recurrence relation G n G n G n where the initial ters G 0 G need to be specified. In this paper we focus on the two ost popular ebers of this faily: The Fibonacci nubers F n Lucas nubers L n which are defined by initial conditions F 0 0 F L 0 2 L respectively. The Binet fors are given by F n αn β n α β L n α n β n n 0. where α β are roots of the quadratic equation x 2 x 0 i.e. α 2 β 2. Disclaier: Stateents conclusions ade in this article are entirely those of the author. They do not necessarily reflect the views of LBBW. 04

2 The goal of this study is to evaluate special finite infinite reciprocal sus of these nubers. The interest is not new soe open probles still exist. For instance no siple expressions for the sus F i L i i F i i are known. The suation of reciprocals of generalized Fibonacci nubers is a challenging issue. It is discussed in several articles where different approaches are applied. The ain lines of research are: establishing algebraic relationships i.e. reduction forulas for these sus expressing the sus in ters of special functions a direct evaluation. Reduction forulas ay be found in the articles [] [6] [7] [6] [7] or [23]. Expressions involving Elliptic functions the Labert series /or Theta functions are derived in [] [2] [4] [] [2] or [3]. A direct evaluation of reciprocal Fibonacci sus is also possible in any cases. Consequently any articles covering the topic exist. Two classical results are contained in the articles [3] [0] where special failies of sus are evaluated exactly. Focusing on reciprocal sus with three ore factors we refer to the research work of Melha. In a series of papers starting in 2000 he gives closed fors for any types of these sus see [4] [] [9] [20] [2]. However as far as we can say the sus that we study here have not been considered by the author. The derivation of the results contained in the present study is based on the following results which we state as theores. Theore.. Let n 0 be integers. Especially i F in F in i L in L in i F in F in i L in L in L i F F n n.2 F L n L Nn..3 F F n α n.4.. F L n αn These expressions are a consequence of the ain result in [2]. Other proofs can be found in [9] [9]. The special case where is even also appears in [8]. The subcase n 0 is presented in [22]. The second theore that is relevant for our study contains findings fro [8]: Theore.2. Let n be integers. i FN F in F in F n F 2 n 0 n F.6 F n

3 Especially i L in L in F n F 2 2 L n L L n L N L Nn L N.7 L Nn i FN F..8 L i L i 2F 2 L N L N L i 2 F in F in F n F 2 α F.9 n F n i L in L in i L i L i F n F 2 2 L n L 2.0 L n α n.. F 2 2L 2 Equation.8 ay be seen as a special case of.7 for n 0. The forulas.2. will be treated as basic fors. These fors ay be cobined to produce ore results of this nature. Specifically soe cobinations produce exact evaluations of reciprocal sus with three factors. In the next section we present applications of the idea. 2 Results We require the following identities: Lea 2.. Let u v be integers. F uv v F u F v F u F v 2. L uv v L u F v L u F v. 2.2 If u v u v have equal parity then { L u F v if v is even F uv F uv 2.3 F u L v if v is odd { F u F v if v is even L uv L uv L u L v if v is odd. Proof. The first two equations can be proved using the relations 2.4 F ab F a F b F a F b L ab L a F b F b L a together with F n n F n equation 2. is given in [8]. The last two stateents follow fro the Binet fors. We oit the details. 06

4 Now we state the ain results of our investigation. Each finding is presented as a separate proposition. Proposition 2.2. Let n be any integers. Define the sus T n N T 2 n N as T n N T 2 n N T n N T 2 n N i i F FN F nf F 2 n n F F FN F nf F 2 n n F F FN F nf F 2 n n F F FN F nf F 2 n n F F in F in F in F in 2. F in F in F in F in. 2.6 F n F n F n F n Proof. We cobine equations.2.6 calculate i F 2 F nn F 2 F n n F 2 F nn F 2 F n n odd. odd F in F in. 2.9 F in F in F in Fro Lea 2. we have F in F F F in F in F in F in F F F in odd. 2.0 Inserting this relation into the above su siplifying using the RHS of.2.6 respectively proves the first stateent. To establish the second forula observe that by definition of the Fibonacci nubers we can write F in F F F in F in F in 2. F in F F F in odd. Corollary 2.3. T n F 2 F nf F 2 F α n F n F 2 F nf F 2 F α n F n F 2 F n α n F 2 F nα n odd 2.2 T 2 n F 2 F nf F 2 F α n F n F 2 F nf F 2 F α n F n F 2 F n α n F 2 F nα n odd

5 Proposition 2.4. Let n be an integer. Let further be even such that /2 is even i.e. 4. Define L i/2n T 3 n N. 2.4 F in F in F in Siilarly for with /2 odd define T 4 n N F i/2n F in F in F in. 2. T 3 n N FN F F N 2.6 F /2 F n F 2 n n F n F F n n T 4 n N FN F F N. 2.7 L /2 F n F 2 n n F n F F n n Proof. The stateents also follow fro Lea 2.: { L in/2 F /2 if is even F in F in 2 F in/2 L /2 if is odd Corollary 2.. With n fro above we have T 3 n 2 F /2 F n F 2 α n F F n 2.9 F F n α n T 4 n 2 L /2 F n F 2 α n F F n F F n α n Reark 2.6. It is obvious that T n N T 2 n N T 4 2 n N T 2 2 n N. Also since L 4i2n F 4in F 4in the expression for T 3 4 n N equals a siple difference of the two basic fors too. To illustrate the results obtained so far we give explicit exaples: T 2 N F 2i2 2 F2N2 F 2i F 2i F 2i3 3 F 2N3 F 2N F 2N F 2N 2 F 2N T 3 3 N T 2 3 6α i F 3i4 3 F3N3 F 3N F 3N 3 F 3i F 3i3 F 3i6 32 F 3N6 F 3N3 32 F 3N

6 T N T 2 3 N T α F 2i F2N2 F 2i F 2i2 F 2i4 3 F 2N4 i F 3i F 3i2 F 3i F 3i4 6 F 2N F 2N F 2N2 3 F 2N T α F3N3 F 3N4 F 3N F 3N F 3N 2 F 3N T 2 3 8α The next propositions contain analogous results for reciprocal Lucas sus. Proposition 2.7. Let n be any integers. Define the sus T n N T 6 n N as T n N i L in L in L in L in 2.29 T n N T 6 n N T 6 n N i F 2 F nf F 2 L n L L n L N L Nn L N F 2 F nf F 2 L n L L n L N L Nn L N L in L in L in L in L Nn L Nn F 2 F nf F 2 L n L L n L N L Nn L N F 2 F nf F 2 L n L L n L N L Nn L N L Nn L Nn F 2 L n L Nn F 2 L nl Nn odd. 2.3 F 2 L n L Nn F 2 L nl Nn odd Proof. Cobine equations.3.7. Fro equation 2.2 of Lea 2. we have { L in F F L in L in L in L in F F L in odd Inserting this relation into the above su siplifying using the RHS of.3.7 respectively proves the first forula. The second forula is proved siilarly. Corollary 2.8. T n F 2 F nf F 2 L n L L n 2 α n F 2 F nf F 2 L n L L n 2 α n F 2 L n α n F 2 L n α n odd

7 T 6 n F 2 F nf F 2 L n L L n 2 α n F 2 F nf F 2 L n L L n 2 α n F 2 L n α n F 2 L n α n odd. 2.3 Proposition 2.9. Let n be an integer. Let further be even such that /2 is even i.e. 4. Define F i/2n T 7 n N L in L in L in Siilarly for with /2 odd define T 8 n N L i/2n L in L in L in T 7 n N L L N L N 2F n F /2 F 2 L n L n L Nn L Nn F /2 F L n L Nn T 8 n N L L N L N. F n F 2 L /2 L n L n L Nn L Nn F L /2 L n L Nn Proof. The stateent is a direct consequence of equation 2.4 of Lea 2.. Corollary T 7 n L 2 2F n F /2 F 2 L n L n α n 2.40 F /2 F L 2 αn T 8 n F n F 2 L /2 2 L n L 2 L n α n. 2.4 F L /2 L 2 αn Reark 2.. It is obvious that T 8 2 n N T 6 2 n N. Finally we state the results in case n 0: Proposition 2.2. Let be an integer. Define the sus T 9 N T 0 N as T 9 N T 0 N i i 0 L i L i L i L i 2.42 L i L i L i L i. 2.43

8 T 9 N F 2F F 2 FN L N L N F 2F F 2 FN L N L N F L F L F 2 LL N F 2 L L N odd 2.44 T 0 N F 2F F 2 FN L N L N F 2F F 2 FN L N L N F L F 2 L L N F F L N odd. 2.4 F 2 LL N Proof. Cobine equations.3.8. The proof is essentially the sae as done above is oitted. Corollary 2.3. T 9 F F F2 2L 2 F F F2 2L 2 F 2 L α F 2 L α odd 2.46 T 0 F F F2 2L 2 F F F2 2L 2 F 2 L α F 2 L α Proposition 2.4. Let be even such that /2 is even i.e. 4. Define odd T N Siilarly for with /2 odd define F i/2 L i L i L i T 2 N L i/2 L i L i L i T N FN F F N 2.0 F /2 2F 2 L N L N L F L L N T 2 N FN F F N. 2. L /2 2F 2 L N L N L F L L N Proof. The stateent is a direct consequence of equation 2.4 of Lea 2.. Corollary 2.. T F /2 F2 2L 2 T 2 L /2 F2 2L 2 F L α 2.2 F L α. 2.3

9 Reark 2.6. For 2 we have that T 2 2 N T 0 2 N. Proposition 2.7. Let be even such that /2 is even i.e. 4. Define T 3 N Siilarly for with /2 odd define T 3 N T 4 N F i/2 L i L i L i L i/2 L i L i L i2. 2. FN F F N 2.6 F 3/2 2F 2 L N L N L F L 2 L N2 T 4 N FN F F N. 2.7 L 3/2 2F 2 L N L N L F L 2 L N2 Proof. Set n in equation.3 cobine with equation.8. Use equation 2.4 of Lea 2. with u i /2 v 3/2. Details are oitted. Corollary 2.8. T 3 F 3/2 F2 2L 2 T 4 L 3/2 F2 Finally we present the following expressions: 2L 2 Proposition 2.9. Let be even such that /2 is even i.e. 4. Define F L 2 α F L 2 α T N Siilarly for with /2 odd define F i/2 L i L i L i T 6 N L i/2 L i L i L i. 2.6 T N FN F F /2 2L F L L N L N L N 2F 2 L N L N L T 6 N L /2 2L 2.62 FN F. F L L N L N L N 2F 2 L N L N L

10 Proof. Set n 0 in.3 cobine with.8. Corollary T F /2 0F 2L 2 F T L /2 0F 2L 2 F 2 Reark 2.2. For 2 we have that T 6 2 N T 9 2 N. We conclude this section with a short list of exaples that are contained in the presentation as special cases: T 6 2 N L 2i L 2i L 2i L 2i3 60 L2N L 2N2 L 2N L 2N3 4 F 2N L 2N T 9 3 N T 9 2 N T α L 2i F2N2 F 2N F 2N L 2i2 L 2i L 2i2 3 L 2N2 L 2N 3 L 2N T i L 3i 3 F3N3 F 3N L 3i3 L 3i L 3i3 32 L 3N3 L 3N 6 F 3N L 3N T F 4i2 T 4 N F4N4 F 4N F 4N L 4i4 L 4i L 4i4 20 L 4N4 L 4N 0 L 4N T F 4i2 T 4 N 8 L 4i4 L 4i L 4i4 490 F4N F4N4 F 4N 2L 4N4 L 4N L 4N4 20 L 4N4 L 4N 2.74 T

11 3 Sus with four factors It is worth to ention that the forulas derived in this article ay be cobined once ore tie to produce closed-for expressions for sus with four factors. We conclude with soe explicit exaples. In all exaples we assue to be an even integer. Let us define the following series: G N G 2 N G 3 N G 4 N F i/2 F i L i L i L i L i2 /2 even 3. L i/2 F i L i L i L i L i2 /2 odd 3.2 F i/2 L i L i L i L i L i2 /2 even 3.3 F i/2 L i L i L i L i L i2 /2 even. 3.4 the following results hold: Corollary 3.. G N T N T 3 N 3. F G 2 N T 6 N T 4 N 3.6 F G 3 N T N F 2 T 3 N 3.7 F 2 G 4 N T N F 2 T 3 N. 3.8 F 2 These results are also valid for N. Proof. Cobine the series fro the previous section use Lea 2.. References [] Alkvist G. 986 A Solution to a Tantalizing Proble The Fibonacci Quarterly [2] André-Jeannin R. 990 Labert Series the Suation of Reciprocals of Certain Fibonacci-Lucas-Type Sequences The Fibonacci Quarterly Vol [3] Backstro R.P 98On Reciprocal Series Related to Fibonacci Nubers with Subscripts in Arithetic Progression The Fibonacci Quarterly

12 [4] Borwein J.M. & Borwein P.B. 987 Pi the AGM John Wiley & Sons New York. [] Brousseau B.A. 969 Suation of Infinite Fibonacci Series The Fibonacci Quarterly [6] Carlitz L. 97 Reduction Forulas for Fibonacci Suations The Fibonacci Quarterly [7] Frontczak R. 206 A Note on a Faily of Alternating Fibonacci Sus Notes on Nuber Theory Discrete Matheatics [8] Frontczak R. 207 Further Results on Arctangent Sus with Applications to Generalized Fibonacci Nubers Notes on Nuber Theory Discrete Matheatics [9] Frontczak R. 207 Suation of Soe Finite Fibonacci Lucas Series unpublished anuscript. [0] Good I. J. 974 A reciprocal series of Fibonacci nubers The Fibonacci Quarterly [] Horada A. F. 988 Elliptic Functions Labert Series in the Suation of Reciprocals in Certain Recurrence-Generated Sequences The Fibonacci Quarterly [2] Hu H. Sun Z.-W. & Liu J.-X. 200 Reciprocal sus of second-order recurrent sequences The Fibonacci Quarterly [3] Melha R. S. 999 Labert Series Elliptic Functions Certain Reciprocal Sus The Fibonacci Quarterly [4] Melha R. S Suation of reciprocals which involve products of ters fro generalized Fibonacci sequences The Fibonacci Quarterly [] Melha R. S. 200 Suation of reciprocals which involve products of ters fro generalized Fibonacci sequences - Part II The Fibonacci Quarterly [6] Melha R. S Reduction Forulas for the Suation of Reciprocals in Certain Second-Order Recurring Sequences The Fibonacci Quarterly [7] Melha R. S On Soe Reciprocal Sus of Brousseau: An Alternative Approach to that of Carlitz The Fibonacci Quarterly [8] Melha R. S. 202 On finite sus of Good Shar that involve reciprocals of Fibonacci nubers Integers Vol. 2 A6. [9] Melha R. S.203 Finite Sus That Involve Reciprocals of Products of Generalized Fibonacci Nubers Integers Vol. 3 A40. [20] Melha R. S. 204 More on Finite Sus that Involve Reciprocals of Products of Generalized Fibonacci Nubers Integers Vol. 4 A4.

13 [2] Melha R. S. 20 On Certain Failies of Finite Reciprocal Sus that Involve Generalized Fibonacci Nubers The Fibonacci Quarterly [22] Melha R. S. & Shannon A. G. 99 On Reciprocal Sus of Chebyshev Related Sequences The Fibonacci Quarterly [23] Rabinowitz S. 999 Algorithic Suation of Reciprocals of Products of Fibonacci Nubers The Fibonacci Quarterly